Use substitution to determine whether the given x-value is a solution of the equation.
$\tan 2x=-\frac{\sqrt{3}}{3},x=\frac{5\pi }{12}$

Verified step by step guidance

1. First, substitute the given value of \(x = \frac{5\pi}{12}\) into the left side of the equation \(\tan 2x\). This means calculating \(\tan\left(2 \times \frac{5\pi}{12}\right)\).
2. Simplify the expression inside the tangent function: \(2 \times \frac{5\pi}{12} = \frac{10\pi}{12} = \frac{5\pi}{6}\), so you need to find \(\tan\left(\frac{5\pi}{6}\right)\).
3. Recall or use the unit circle to find the exact value of \(\tan\left(\frac{5\pi}{6}\right)\). Remember that \(\frac{5\pi}{6}\) is in the second quadrant where tangent is negative.
4. Next, evaluate the right side of the equation, which is \(-\frac{\sqrt{3}}{3}\). This is a constant value you can compare with the left side.
5. Finally, compare the value of \(\tan\left(\frac{5\pi}{6}\right)\) with \(-\frac{\sqrt{3}}{3}\). If they are equal, then \(x = \frac{5\pi}{12}\) is a solution; if not, it is not a solution.

Key Concepts

Tangent Function and Its Properties

The tangent function, tan(θ), is defined as the ratio of sine to cosine (sin θ / cos θ). It is periodic with period π, meaning tan(θ + π) = tan(θ). Understanding how to evaluate tangent at specific angles, especially multiples of π, is essential for solving equations involving tan(2x).

Substitution Method in Trigonometric Equations

Substitution involves replacing the variable x with a given value to verify if it satisfies the equation. Here, substituting x = 5π/12 into tan(2x) allows direct evaluation to check if it equals the given expression. This method helps confirm whether the proposed x-value is a solution.

Simplifying and Evaluating Trigonometric Expressions

Evaluating tan(2x) at x = 5π/12 requires simplifying the angle 2x = 5π/6 and then calculating tan(5π/6). Knowing exact values of tangent at standard angles and simplifying radicals like √3/3 is crucial to compare both sides of the equation accurately.